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A226206
Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles of area > 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
10
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 3, 1, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 5, 0, 7, 0, 5, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 7, 7, 0, 0, 0, 0, 0, 1
OFFSET
0,41
LINKS
EXAMPLE
A(6,4) = A(4,6) = 3:
._._._._._._. ._._._._._._. ._._._._._._.
| | | | | | | | | |
|___|___|___| | |___| |___| |
| | | | | | | | | |
|___|___|___| |_______|___| |___|_______| .
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, ...
1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, ...
1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, ...
1, 0, 1, 1, 3, 2, 7, 7, 16, 19, 40, ...
1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 2, ...
1, 0, 1, 0, 5, 0, 16, 0, 48, 0, 160, ...
1, 0, 0, 1, 0, 0, 19, 0, 0, 50, 17, ...
1, 0, 1, 0, 8, 1, 40, 2, 160, 17, 796, ...
...
MAPLE
b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then 0 elif n=0 or l=[] then 1
elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
else for k do if l[k]=0 then break fi od; s:=0;
for i from k+1 to nops(l) while l[i]=0 do s:=s+
b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
od; s
fi
end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which [Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = 0; For[i = k+1, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join [l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]] ]]]; s]]; a [n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)
CROSSREFS
Main diagonal gives A347800.
Cf. A219924.
Sequence in context: A364042 A033764 A033784 * A350682 A375150 A319660
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 31 2013
STATUS
approved